It has been long hypothesized that the structure of a material bombarded by energetic particles might approach a certain asymptotic steady state in the limit of high exposure to irradiation. There is still no definitive verdict regarding the validity of this hypothesis or the conditions where it applies. To clarify this, we explore a highly simplified model for microstructural evolution that retains full atomic detail of the underlying crystal structure and involves random events of generation and relaxation of defects. We explore the dynamics of evolution of the model in the limit T = 0, where the defect and dislocation microstructure is driven purely by the spatially fluctuating stress field accumulating as a result of stochastic generation of point defects. Using body-centred cubic iron and tungsten as examples, we show that microstructure exhibits a structural transition and then approaches a limiting asymptotic state at doses of order O(0.1) and O(1) canonical defects per atom, respectively, and analyse the microscopic and macroscopic parameters characterizing both the transition and the asymptotic microstructural state.